For questions conserning circles. A circle is a curve composed of points in a plane that are at a fixed distance from a fixed point.

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Finding circle with two points on it and a tangent from one of the points

Two points P1(x1,y1) and P2(x2,y2) are known. In addition, a line slope passing through P1 is known. The aim is to construct a circle (or circular arc) that it passes through both P1 and P2 and it is ...
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96 views

Differentiation of a circle

As a discus thrower is spinning counterclockwise to throw a discus, the discus travels along the path given by the circle $x^2+y^2=4$. If the discus is released at the point $(\sqrt2,\sqrt2)$ and ...
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2answers
144 views

Beautiful triangle problem

Circle, inscribed in $ABC$, touches $BC, CA, AB$ in points $A', B', C'$. $AA' BB', CC'$ intersect at $G$. Circumcircle of $GA'B'$ crosses the second time lines $AC$ and $BC$ at $C_A$ and $C_B$. Points ...
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1answer
66 views

In every polygon circumscribed about a circle, there exist three sides that can form a triangle.

How can one show that in every polygon circumscribed about a circle, there exist three sides that can form a triangle? (This was posted by another user and then deleted while I was typing my answer.) ...
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Comparing The Rates at Which Squares and Circles Fill Large Similar Areas.

Consider these two search patterns. ${\square}$ A square moves in straight lines forming what you might call a "square-spiral" pattern as it covers a much larger square space. ${\bigcirc}$ A circle ...
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134 views

Number of ways to seat people around a circular table

I got (i) which is $9!$, but there are no answers for the second question. I stated that $$P(\text{none together})=1-P(\text{3 together})-P(\text{2 together})$$ and got the answer $7/12$. Is this ...
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5answers
414 views

How can I find the radius of a circle from a chord and a section of the radius?

Draw a circle with center O. Line AD is a chord that is 8cm long. The arc above is smaller than the one below. B is the center of AD. Line CB is a line that is 2cm long. It meets AD at 90°. ...
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1answer
87 views

Computing overlapping circle positions, equidistant from each other.

Hello, I am a programmer and I wanted to develop an application that would have several overlapping circles in the same location, where each circle can be selectable. Is there a mathematical way of ...
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0answers
119 views

Find the minimum radius of the circle which is orthogonal to two given circles

Problem : Find the minimum radius of the circle which is orthogonal to both the circles $x^2+y^2-12x+35=0$ and $x^2+y^2+4x+3=0$ . Solution : Let the equations : $x^2+y^2-12x+35=0.....(i)$ and ...
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3answers
44 views

Cyclic quadrilaterals - finding the size of an angle

I know this might seem like a really simple question, but I really don't understand where I am going wrong. I am familiar with cyclic quadrilaterals as well as their properties, but this question ...
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3answers
46 views

Find the radii of the two circles which pass through the point $(16,2)$ and touch both axes

How can I find the radii of the two circles which pass through the point $(16,2)$ and touch both the axes? I've only ever seen demonstrations using three normal co-ordinates; or two normal ...
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2answers
426 views

If the length of tangent drawn from an external point P to the circle of radius $r$ is $l$ , then prove that area…

Problem : If the length of tangent drawn from an external point P to the circle of radius $r$ is $l$ , then prove that area of triangle form by pair of tangent and its chord of contact is ...
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1answer
265 views

Areas between intersecting chords

In the circle below let the two chords be called $C_1$ and $C_2$, and their intersection be some point that is not the center. The chord power theorem tell us that $a \cdot b = c \cdot d$. I am ...
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1answer
244 views

How to calculate point on circumference of circle given radius

I am trying to come up with a formula to calculate the y co-ordinate of the point on the circle in the attached picture (i.e. delta y) based on the circumference of the circle and the distance x. ...
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0answers
54 views

Inverse with respect to a given circle

Determine the inverse with respect to a given circle $g:\mathbb{R}^{2} \to \mathbb{R}^{+}, g(x,y)=x^{2}+y^{2}$. I have looked around for non geometric derivations without finding any of value. ...
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2answers
436 views

Find the maximum perpendicular height between a chord and an arc.

I am doing a maths modelling project, and I am stuck on a part. I have a (arc length) and L (chord length), but I want to find H, the maximum perpendicular distance between them! Any help would be ...
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1answer
160 views

Show that the common tangents to circles $x^2+y^2+2x=0$ and $x^2+y^2-6x=0$ …

Problem : Show that the common tangents to circles $x^2+y^2+2x=0$ and $x^2+y^2-6x=0$ form an equilateral triangle. Solution : Let $C_1 : x^2+y^2+2x=0$ here centre of the circle is $(-1,0) $ and ...
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4answers
254 views

Equation of a line tangent to circumference

Discover the general equation of the tangent line to the circumference $x^2 + y^2 - 2x + 4y + 1 = 0$ by the point $(3,4)$. NO CALCULUS. by the circumference equation i discovered that $C(1, ...
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1answer
73 views

intersection of 4 circles

Hi I'm doing some programming challenges for fun and I am trying to work out the maths required to solve this problem. It has been 10 years since I did any maths in anger like this so i'm a bit ...
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1answer
488 views

Infinite staircase to a circle

Suppose you start at $(0,0)$ on the unit disc and repeat the following procedure again and again: Face east and walk half-way to the circumference. Face north and walk half-way to the circumference. ...
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1answer
55 views

Circle Equation Surjectivity

Consider the circular function $g:\mathbb{R}^{2} \to \mathbb{R}^{+}$, $g(x,y)=x^{2}+y^{2}$. Show that it is surjective and continuous. Note This post has been amended in accordance with the ...
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3answers
868 views

How to determine family of circles passing through two given points?

The question asks to show that the equation of any circle passing through two given points takes a certain form. I have obtained the points as being $(2,1)$ and $(2,-1)$ but I'm not sure as to how to ...
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1answer
79 views

power of a point (circles) questions.

Lets say we have two circles call them $O_{1}$ and $O_{2}$. Let $a_{1}$ and $a_{2}$ be the arcs of the circles. Then when it comes to two circles, three cases arise. They intersect at two points, they ...
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2answers
173 views

Show that four vertices of a square cannot lie on four concentric circles, radii of which form an arithmetic sequence

My teacher said it's solved using proof through contradiction. I've considered cases of the centre of the circle, but I lose geometry big time so not sure how to do this.
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144 views

Symmetrical of a triangle's vertexes

I have the following problem : Show that the symmetrical (ie reflection) of a triangle's vertexes by the opposite side are aligned iff the distance between the orthocenter and the circumcenter is ...
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1answer
138 views

Quadrilateral Inscribed angles calculation with one arc angle

I am trying desperately to solve following problem. How can I solve it, the image and question is included in image
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2answers
576 views

A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and int…

Problem : A circle touches the parabola $y^2=4ax$ at P. It also passes through the focus S of the parabola and intersects its axis at Q. If angle SPQ is $\frac{\pi}{2}$ find the equation of the ...
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1answer
161 views

Using an offset data point with x, y coords to find the true centre of a circle

I have a data point at (0, 0) where measurements of a tank's shell are taken from. I have used this data point to plot the circle in a graph. However, this data point is not the true centre of the ...
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4answers
1k views

Area of the intersection of four circles of equal radius [duplicate]

This picture basically shows a rearrangement of four quarters of a circle of radius 1. It asks for the shaded area. I got the answer to be $\frac{2\pi + 6}{13}$. But then it is incorrect. The way ...
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2answers
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Task “Inversion” (geometry with many circles)

Incircle $\omega$ of triangle $ABC$ with center in point $I$ touches $AB, BC, CA$ in points $C_{1}, A_{1}, B_{1}$. Сircumcircle of triangle $AB_{1}C_{1}$ intersects second time circumcircle of $ABC$ ...
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1answer
55 views

Average rate of speed relative to a given point

For this question I am mainly concerned about points A and B on the image below and the image below hopefully helps illustrate my question. If point B is fixed and A has to move in a strait line in ...
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1answer
116 views

Circular variation with repetition

I would like to know formula for circular variation with repetition. What I mean is : You have round table with n-spots. On every spot there can be number from 1 to k. So for n = 4 and k = 3 ...
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1answer
77 views

find the area of value of b in the equilateral

A circle meets the sides of an equilateral triangle ABC at six points D, E, F ,G, H , I in the figure . If AE= 4 ED = 26 , FG = 14 , and the circle with diameter HI has area πb, find b. sorry i ...
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1answer
123 views

In the circle below , mA= 86, mBDC= 32, mAD= 48 find the mBC, mCD

In the circle below, m∠A=86, m∠BDC=32, and mA͡D= 48 find mB͡C, mC͡D, mA͡B, m∠ADB, m∠ABD, m∠DBC, m∠BCD
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1answer
45 views

Finding the number of Circle or Circles in a Circle

Let a circle $A$ which radius is $10 m$ and another circle is $B$ which radius is $0.2 m$.Is it possible to say that what is the maximum number of circles $B$ can be drawn in circle $A$? I tried much ...
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2answers
138 views

High-School level question concerning circle and arcs

This question somehow is unsolvable to me. Any idead/hints wil be much appreciated. $AB$ is a chord which is cut ny the chords $CD$ and $EC$ in the circle. Givens: $\frown{AC} ...
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1answer
25 views

No four points with pairwise distance 1 can be contained inside a halfdisk of radius 1.

An open disk $D$ of radius $1$ in the Euclidean plane is the set of points with distance less than $1$ to the center of the disk. An open half disk $H$ of radius $1$ is obtained by "cutting" $D$ ...
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Proving a trigonometric relation using circle properties

Hi, I've been having trouble with this question, and would really like some help. What I've done so far is applied the cosine rule in the triangle PQR to find that $PR^2=a^2+c^2+2ac\cos\theta$. ...
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3answers
1k views

Write the equation of the tangent line of a circle

I'm totally lost with this question. I appreciate any kind of help. if the equation of a circle is $(x-3)^2+y^2=9$ Find : -Equation of the tangent line at $(2,2\sqrt2)$ -Equation of the tangent ...
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3answers
943 views

What does the Circle really mean?

Which of the following figure is really the circle? If a point is on the circle it means that point should be on the circumference is it? (point $Z$ on figure 1). Point $P$ on figure $1$ is inside ...
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1answer
74 views

Radius of circumscribed circle of triangle as function of the sides

Given the length ot the sides $a , b$ and $c$ of $ \triangle ABC$. What is the length of the radius of the circumcribed circle? After some formula substitution I came to the monster formula: $$ ...
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Proof: At most 3 circles of radius 1/2 fit into the interior of a halfcircle of radius 1

It is a well known fact that at most 7 interior disjoint circles of radius 1/2 can be centered in a circle of radius 1; note that they don't need to be fully contained in the radius 1 circle. I am ...
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2answers
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I got stucked in middle of the problem. How to find the value of radius 'x' cm from the given figure?

![enter image description here][2] Firstly, I calculated the area of sector $AOB$ by applying $\frac{1}{2}\times (1.2\ \text{radians})\times 20^{2}$ (formula for area of sector of circle) and ...
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1answer
143 views

Geometry question: ray paths and circles

I was working on a problem and used the image below to make an argument regarding an effective line-of-sight (from one of my papers). My question below is more of an intellectual curiosity since the ...
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2answers
148 views

Prove that $\angle BAC + \angle OAP = 180^\circ$

Prove that if you construct two circle centered at O and P and intersecting at A with tangent lines BA and CA. Prove that $\angle BAC + \angle OAP = 180^\circ$. I'm having trouble just starting the ...
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2answers
7k views

Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
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1answer
246 views

How many circles with radius $r_1$ can be inscribed in circle with radius $r_2$

Is there formula for finding the number of inscribed circles in a bigger circle? For example: Little circles radius: $7 cm$; Big circle radius: $50cm$;
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5answers
694 views

Coordinates of the point on the circle inscribed in a square

I try to find a way to calculate coordinates of a point nested on a circle inscribed in a square. The available variables, are: 1) side length of the square = 100; 2) circle radius = 50; 3) angle (a) ...
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249 views

Pre calculus Unit Circle

Suppose that you did not have the Unit Circle on Circle A, but rather a circle of radius $5$. Will the angle measures in degrees and/or radians change? Why or why not? Suppose that you did not have ...
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1answer
119 views

Area of similar triangle

Suppose that we are given a triangle whose area is known. put a circle C of radius r inside that triangle. How can we find the area of a triangle similar to the first one and whose inscribed circle is ...